On Degrees in Multihomogeneous Ideal Theory

The special case of an intersection of an algebraic variety with a hyperplane plays an important role in Bézout’s theorem. In this area J. Stückrad and W. Vogel proved the so-called h1-condition in 1971 (see [4]). We think this condition is also important in the multihomogeous case of Bézout’s theorem and so we give here the extension of this condition for this case. We want also look at the new investigations in [3, 5] of the problem of the converse to Bézout’s theorem. At the end of our paper we discuss a relation, which is used by D. W. Masser and G. Wüstholz in the proof of Lemma A1 in [1]. Here we confine ourselves to the bihomogeneous case, because we obtain all results in the multihomogeneous case analogously. Let R := K[x0, . . . , xk, y0, . . . , ym] be a polynomial ring over an infinite field K. Let a be a 0 ≤ d-dimensional multihomogeneous ideal of R. We let H(s, t; a) denote the Hilbert function of a. For large s and t H(s, t; a) is a polynomial in s and t: